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for general service. On the whole, however, nearly parallel, that it would be ridiculous affecsmall pieces of artillery have been brought into tation to pay any regard to the deviations from use : thus the battering pieces now approved equality and parallelism. A bullet rising a mile are the demi-cannon of former times; it being above the surface of the earth loses only do of found that their stroke, though less violent than its weight, and a horizontal range of four miles that of a larger piece, is yet sufficiently adapted only four of deviation from parallelism. Gravito the strength of the usual profiles of fortifica- tation may be therefore assumed as equal and tion; and that the facility of their carriage and parallel. The errors arising from this assumpmanagement, and the ammunition they spare, tion are quite insensible in all the uses which give them great advantages beyond the whole can be made of this theory; which was the first cannon formerly employed. The method of fruits of mathematical philosophy, and the effort making a breach, by first cutting off the whole of the genius of the great Galileo. 'wall as low as possible before its upper part is Gravity is a constant or uniform accelerating attempted to be beaten down, seems also to be a or retarding force, according as it produces the considerable modern improvement. But the descent, or retards the ascent, of a body: and, most important advance in this art is the method all other forces being ascertained by the acceleof firing with small quantities of powder, and rations which they produce, they are convenientelevating the piece so that the bullet may just goly measured by comparing their accelerations clear of the parapet of the enemy, and drop into with the acceleration of gravity. This therefore his works. By these means the bullet, coming has been assumed by all the latest and best to the ground at a small angle, and with a small writers on mechanical philosophy, as the unit by velocity, does not bury itself, but bounds or rolls which every other force is measured. It gives a along in the direction in which it was fired : and perfectly distinct notion of the force which retherefore, if the piece be placed in a line with tains the moon in its orbit, to say it is the 3600th the battery it is intended to silence, or the front part of the weight of the moon at the surface of it is to sweep, each shot rakes the whole length the earth : i. e. if a bullet were here weighed by of that battery or front; and has thereby a much a spring steel-yard, and pulled it out to the mark greater chance of disabling the defendants, and 3600, if it were then taken to the distance of the dismounting their cannon. This method was in- moon, it would pull it out only to the mark 1. vented by Vauban, and was by him styled Bat- This assertion is made from observing that a terie á Ricochet. It was first practised in 1692 body at the distance of the moon falls from that at the siege of Aeth. Something similar was prac- distance robo part of sixteen feet in a second. tised by the king of Prussia at the battle of Ros- Forces therefore which are imperceptible are not bach. in 1757.
compared, but the accelerations, which are their
indications, effects, and measures. For this reaPART I.
son philosophers have been anxious to determine THEORY OF PROJECTILES. with precision the fall of heavy bodies, to have
an exact value of the accelerating power of terSect. I.—Of The Effects of Gravity on Pro- restrial gravity. This measure may be taken in JECTED BODIES.
tww ways; by taking the space through which It has been demonstrated that a body pro the heavy body falls in a second; or the velocity jected in the usual way from the surface of the which it acquires in consequence of gravity earth in the atmosphere, must describe a conic having acted on it during a second. The last is section, having the centre of the earth in one the proper measure; for the last is the immefocus, and that it will describe round that focus diate effect on the body. The action of gravity areas proportional to the times : it follows that, has changed the state of the body, by giving it a if the velocity of projection exceeds 36,700 feet determination to motion downward : this both in a second, the body (if not resisted by the air) points out the kind and the degree or intensity would describe a hyperbola ; if it be just 36,700 of the force of gravity. The space described it would describe a parabola ; and, if it be less, in a second by falling is not an invariable meait would describe an ellipsis. If projected di- sure; for, in the successive seconds, the body rectly upwards, in the first case, it would never falls through 16, 48, 80, 112, &c., feet, but the return, but proceed for ever; its velocity con- changes of the body's state in each second is the tinually diminishing, but never becoming less At the beginning it had no determinathan an assignable portion of the excess of the tion to move with any appreciable velocity; at initial velocity above 36,700 feet in a second; in the end of the first second it had a determination the second case it would never return, its velo- by which it would have gone on for ever (had no city would diminish without end, but never be subsequent force acted on it) at the rate of extinguished. In the third case, it would proceed thirty-two feet per second. At the end of the till its velocity was reduced 10 an assignable por- second second, it had a determination by which tion of the difference between 36700 and its it would have moved for ever, at the rate of initial velocity; and would then return, regain- sixty-four feet per second. At the end of the ing its velocity by the same degrees, and in the third second, it had a determination by which same places as it lost it. These are necessary it would have moved for 'ever, at the rate of consequences of a gravity directed to the centre ninety-six feet per second, &c. &c. The differof the earth, and inversely proportional to the ence of these determinations is a determination square of the distance. But, in the greatest pro- to the rate of thirty-two feet per second. This jections that we are able make, the gravita- is therefore constant, and the indication and tions are so nearly equal, and in directions so proper measure of the constant or invariable
force of gravity. The space fallen through in and therefore its whole velocity will be 138 feet the first second is of use only as it is one-half of per second. the measure of this determination; and, as halves 6. To find how far it will have moved, comhave the proportion of their wholes, different pound its motion of projection, which will be accelerating forces may be safely affirmed to be forty feet in four seconds, with the motion which in the proportion of the spaces through which gravity alone would have given it in that time, they uniformly impel bodies in the same time. which is 256 feet; and the whole motion will be But we must always recollect that this is but 296 feet. one-half of the true measure of the accelerating 7. Suppose the body projected as already force. Mathematicians of the first rank have com- mentioned, and that it is required to determine mitted great mistakes by not attending to this; and the time it will take to go 296 feet downwards, it is necessary to notice it here, because cases will and the velocity it will have acquired. Find the occur, in the prosecution of this subject, where height x, through which it must fall to acquire we shall be very apt to confound our reasovings the velocity of projection, ten feet, and the time by a confusion in the use of those measures. y of falling from this height. Then find the time Sect. II.-OF THE MEASURE OF THE ACCELE
z of falling through the height 296 + x, and the RATIVE POWER OF GRAVITY.
velocity v acquired by this fall. The time of
describing the 296 feet will be z - y, and •v is The accurate measure of the accelerative power the velocity required. From such examples it of gravity is the fall 1614 feet, if measured by is easy to see the way of answering every questhe space, or the velocity of 32 feet per second, tion of the kind. if the velocity be taken. It will greatly facilitate Writers on the higher parts of mechanics alcalculation, and will be sufficiently exact for ways compute the actions of other accelerating every purpose, to take 16 and 32, supposing and retarding forces, by comparing them with that a body falls sixteen feet in a second, and the acceleration of gravity; and, to render their acquires the velocity of thirty-two feet per second. expressions more general, use a symbol, such as Then, because the heights are as the squares of g for gravity, leaving the reader to convert it the times, and as the squares of the acquired ve into numbers. Agreeably to this view, the gelocities, a body will fall one foot in one fourth of neral formulæ will stand thus : a second, and will acquire the velocity of eight
I. v = 2 g h, i. e. M 2 M g / n,
gt, feet per second. Let h express the height in feet, and call it the producing height; v the ve
4 h ✓
II. t= locity in feet per second, and call it the produced
✓ 2 g
8 velocity, the velocity due; and t the time in
qa seconds.—The following formulæ, which are of III. h=
2 easy recullection, will serve, without tables, to
2 g answer all questions relative to projectiles.
Gravity, or its accelerating power, is estimated I. v=8nh, = 8 X 4t, = 32 t
in all these equations, as it ought to be, by the
change of velocity which it generates in a parII. t =
ticle of matter in a unit of time. 32
mathematicians, in their investigations of curviIII. vh= * = 41
lineal and other varied motions, measure it by
the deflection which it produces in this time from 22 IV. h= = 16 (2
the tangent of the curve, or by the increment by 64
which the space described in a unit of time exTo give some examples of their use, let it be ceeds the space described in the preceding unit. required,
This is but one-half of the increment which grai. To find the time of falling through 256 feet. vity would have produced, had the body moved
16 Here h = 256, ✓ 256 = 16, and
through the whole moment with the acquired
ă =4. An- addition of velocity. In this sense of the symbol swer 4'.
g, the equations stand thus : 2. To find the velocity acquired by falling four I. p=2 g , = 2 g seconds. t=4:32 X 4= 128 feet per second. II. t=
3. To find the velocity acquired by falling 625 feet. h= 625. vh=25- 8 Nh= 200 feet
, and wh=2V & 4. To find the height to which a body will rise
4 g' when projected with the velocity of fifty-six feet It is likewise very common to consider the per second, or the height through which a body accelerating force of gravity as the unit of commust fall to acquire this velocity.
parison. This renders the expressions much 56
more simple. In this way v expresses not the u= 56 = 7,=»h.7 = h, = 49 feet. 8
velocity, but the height necessary for acquiring 3136
it, and the velocity itself is expressed by ✓ v. or 56% = 3136 · = 49 feet. 64
To reduce such an expression of a velocity to 5. Suppose a body projected directly down- numbers, multiply it by ✓ 2g, or by 2 v g. wards with the velocity of ten feet per second; according as g is the generated velocity, or the what will be its velocity after four seconds? In space fallen through in the unit of time. This four seconds it will have acquired, by the action will suffice for the perpendicular ascents or deof gravity, the velocity of 4 x 32, or 128 feet, scents of heavy bodies; and we proceed to con
sider their motions when projected obliquely. 4. The times of describing the different arches The circumstance which renders this an interest- BV, VG, of the parabola are as the portions BC, ing subject is, that the flight of cannon shot and BH, of the tangent, or as the portions A D, Ad, shells are instances of such motion, and the art of the directrix, intercepted by the same vertical of gunnery must in a great measure depend on lines AB, CV, HG; for the times of describing this doctrine. Let a body
BV,BVG, are the same with those of describing B (fig. 1) be projected
the corresponding parts B C, BH, of the tangent,
H н in any direction, BC, not
and are proportional to these parts, because the perpendicular to the ho
motion along BH is uniform; and BH, are rizon, and with any ve
proportional to AD, Ad. Therefore the motion locity. Let A B be the .
estimated horizontally is uniform. height producing this
5. The velocity in any point G of the curve is velocity; that is, let the
the same with that which a heavy body would velocity be that which ALA
acquire by falling from the directrix along dG. a heavy body would ac
Draw the tangent GT, cutting the vertical AB quire by falling freely
in T; take the points a, f, equidistant from A through A B. It is re
and a, and extremely near them, and draw the
gyG quired to determine the B
verticles a b, f g; let the points a, f, continually path of the body, and
approach A and d, and ultimately coincide with all the circumstances of
them. B b will therefore ultimately be to gG its motion in this path?
in the ratio of the velocity at B to the velocity at 1. By the continual
G (for the portions of the tangent ultimately action of gravity, the E
coincide with the portions of the curvey and are body will be continually
described in equal times); but B b is to Gg as deflected from the line
BH to TG: therefore the velocity at B is to that BC, and will describe
at G as BH to TG. But, by the properties of a curve line B VG, con
the parabola, BH' is to TGas AB to dG; and cave towards the earth.
A B is to dG as the square of the velocity ac2. This curve line is
quired by falling through A B to the square of a parabola, of which the vertical line ABE is the velocity acquired by falling through d G; diameter, B the vertex of this diameter, and BC and the velocity in BH, or in the point B of the a tangent in B. Through any two points, VG, parabola, is the velocity acquired by falling along of the curve draw VC, GỦ, parallel to AB, AB; therefore the velocity in TG, or in the meeting BC in C and H, and draw VE, GK, point G of the parabola, is the velocity acquired parallel to BC, meeting AB in E, K. It fol- by falling along d G. lows, from the composition of motions, that the The preceding propositions contain all the body would arrive at the points V, G, of the curve theory of the motion of projectiles in vacuo, or in the same time that it would have uniformly independent on the resistance of the air; and described BC, BH, with the velocity of projec- being a very easy and neat exhibition of mathetion; or that it would have fallen through BE, matical philosophy, and connected with a very BK, with a motion uniformly accelerated by interesting practice, they have been much comgravity; therefore the times of describing BC, mented on, and have furnished matter for many BH, uniformly, are the same with the time of splendid volumes. But the resistance of the air falling through BE, BK. But, because the mo- occasions such a prodigious diminution of motion along B H is uniform, BC is to BH as the tion in the great velocities of military projectiles, time of describing BC to the time of describing that this parabolic theory, as it is called, is of BH, which we may express thus, BC: BH= little practical use. A musket ball, discharged T, BC: T, BH, = T, BE:T, BK. But, be- with the ordinary allotment of powder, issues cause the motion along BK is uniformly acce- from the piece with the velocity of 1670 feet per lerated, we have BE:
BK = T, BE:T, BK, second: this velocity would be acquired by fall= BC : BH”, = EV? : KG?; therefore the ing from the height of eight miles. If the piece curve BVG is such, that the abscissæ BE, BK, be elevated to an angle of 45°, the parabola are as the squares of the corresponding ordinates should be of such extent that it would reach sixEV, KG; that is, the curve BVG is a para- teen miles on the horizontal plain; whereas it bola, and BC, parallel to the ordinates, is a tan- does not reach much above half a mile Simigent in the point B.
lar deficiencies are observed in the ranges of 3. If the horizontal line ADd be drawn cannon shot. It is unnecessary, therefore, to through the point A, it is the directrix of the enlarge upon this theory. parabola. Let BE be taken equal to AB. The Facts prove, beyond all doubt, how deficient time of falling through B E is equal to the time the parabolic theory is, and how unfit for directof falling through AB; but BC is described ing the practice of the artillerist. A very simple with the velocity acquired by falling through consideration is sufficient for rendering this obAB: and therefore by number 4 of perpendicu- vious to the most uninstructed. The resistance lar descents, BC is double of AB, and EV is of the air to a very light body may greatly exceed double of B E; therefore, E Vo = 4 BE, = 4 its weight. Any one will feel this in trying to BE X AB, = BE X 4 A B, and 4 AB is the move a fan very rapidly through the air; thereparameter or latus rectum of the parabola BVG, fore this resistance would occasion a greater deand, A B being one fourth of the parameter, A D viation from uniform motion than gravity would is the directrix.
in that body. Its path, therefore, through the
air may differ more from a parabola than a para- gravity. The weight W of a body is the aggrebola itself deviates from the straight line. For gate of the action of the force of gravity g on these reasons, we affirm that the voluminous each particle of the body. Suppose the number treatises which have been published on this sub- of equal particles, or the quantity of matter, of a ject are nothing but ingenious amusements for body, to be M, then W is equivalent to g M. In young mathematicians. All that seems possible like manner, the resistance R, observed in any to do for the practical artillerist is, to multiply experiment, is the aggregate of the action of a judicious experiments on real pieces of ordnance, retarding force R’ on each particle, and is equiwith the charges that are used in actual service,
W and to furnish him with tables calculated from valent to R’M: and as g is equal to
M' such experiments.
R is equal to
Let us keep this distinction in Sect. III.–Of the Causes of the DEFICIENCY view, by adding the differential mark' to the letOF THE PARABOLIC THEORY.
ter R or r, which expresses the aggregate resistIt is, however, the business of the philosopher ance. to enquire into the causes of such a prodigious If we thus consider resistance as a retarding deviation from a well founded theory; and, hav- force, we can compare it with any other such ing discovered them, to ascertain precisely the force by means of the retardation which it prodeviations they occasion. Thus we shall obtain duces in similar circumstances. We would another theory, either in the form of the parabo- compare it with gravity by comparing the dimilic theory corrected, or as a subject of independ- nution of velocity which its uniform action proent discussion.
duces in a given time with the diminution proThe motion of projectiles being performed in duced in the same time by gravity. But we the atmosphere, the air is displaced, or put in have no opportunity of doing this directly; for, motion. Whatever motion it'acquires must be when the resistance of the air diminishes the vetaken from the bullet. The motion communi- locity of a body, it diminishes it gradually, cated to the air must be in the proportion of the which occasions a gradual diminution of its own quantity of air put in motion, and of the velocity intensity. This is not the case with gravity, communicated to it. If, therefore, the displaced which has the same action on a body in motion air be always similarly displaced, whatever be or at rest. We cannot, therefore, observe the the velocity of the bullet, the motion communi- uniform action of the resistance of the air as a cated to it, and lost by the bullet, must be pro- retarding force. We must make the comparison portional to the square of the velocity of the in some other way. We can state them both as bullet and to the density of the air jointly. dead pressures. A ball may be fitted to the rod Therefore the diminution of its motion must be of a spring steelyard, and exposed to the impulse greater when the motion itself is greater; and in of the wind. This will compress the steelyard to the very great velocity of shot and shells it must the mark 3, for instance. Perhaps the weight of be prodigious. From Mr. Robins's experiments this ball will compress it to the mark 6. Half it is plain that a globe of four inches and a half this weight would compress it to 3. We reckon in diameter, moving with the velocity of twenty- this equal to the pressure of the air, because five feet in a second, sustained a resistance of they balance the same elasticity of the spring. 315 grains, nearly three-quarters of an ounce. In this way we can estimate the resistance by Suppose this ball to move 800 feet in a second, weights whose pressures are equal to its presthat is, thirty-two times faster, its resistance sure; and we can thus compare it with other would be 32 x 32 times three-quarters of an resistances, weights, or any other pressures. In cunce, or 768 ounces, or forty-eight pounds. fact, we are measuring them by all the elasticity This is four times the weight of a ball of cast of the spring. This elasticity in its different iron of this diameter; and, if the initial velocity positions is supposed to have the proportions of had been 1600 feet per second, the resistance the weights which keep it in these positions. would be at least sixteen times the weight of the Thus we reason from the nature of gravity, no ball. It is indeed much greater.
longer considered as a dead pressure, but as a So great a resistance, operating constantly and retarding force; and we apply our conclusions uniformly on the ball, must take away four times to resistances which exhibit the same pressures, as much from its velocity as its gravity would do but which we cannot make to act uniformly: in the same time. In one second gravity would This sense of the words must be remembered reduce the velocity 800 to 768, if the ball were whenever we speak of resistances in pounds and projected straight upwards. This resistance of ounces. the air would therefore reduce it in one second The most convenient and direct way of stating to 672, if it operated uniformly; but as the ve the comparison between the resistance of the air locity diminishes continually by the resistance, and the accelerating force of gravity, is to take a and the resistance diminishes along with the ve case in which we know that they are equal. locity, the real diminution will be somewhat less Since the resistance is here assumed as proporthan 128 feet. We shall, however, find, that in tional to the square of the velocity, it is evident one second its velocity will be reduced from that the velocity may be so increased that the re800 to 687. From this instance it is clear that sistance shall equal or exceed the weight of the the resistance of the air must occasion great de- body. If a body be already moving downwards viation from parabolic motion.
with this velocity, it cannot accelerate; because To judge accurately of its effect, we must the accelerating force of gravity is balanced by consider it as a retarding force, as we consider an equal retarding force of resistance. It follows
from this remark that this velocity is the greatest locities, and the resistance to the velocity iz that a body can acquire by the force of gravity u?
R will be == only. Nay, we shall see that it never can com- 2 a
Moreover, the times in
2a pletely attain it; because, as it aproaches to this which the same velocity will be extinguished by velocity, the remaining accelerating force de- different forces, acting uniformly, are inversely creases faster than the velocity increases. It as the forces, and gravity would extinguish the may therefore be called the limiting or terminal velocity by gravity.
velocity 1 in the time = (in these measures)
8 Let a be the height through which a heavy
1 2 a body must fall, in vacuo, to acquire its terminal to
Therefore we have the following velocity in air. If projected directly upwards with this velocity, it will rise again to this height, and the height is half the space which it would proportion, (=R),(= 5) ) = : 2 a,
u" describe uniformly, with this velocity, in the time and 2 a is equal to E, the time in which the veof its ascent. Therefore, the resistance to this velocity being equal to the weight of the body, tion of the resistance competent to this velocity.
locity 1 will be extinguished by the uniform acit would extinguish this velocity, by its uniform action, in the same time, and after the same dis- The velocity 1 would in this case be extinguishtance, that gravity would. Now leto be the the space described is one-half of what would be
ed after a motion uniformly retarded, in which during an unit of time, and let u be the terminal uniformly described during the same time with velocity of any particular body. The theoremis thus described by a motion which begins with
the constant velocity 1. Therefore the space for perpendicular ascents give us g = the velocity 1, and is uniformly retarded by the
resistance competent to this velocity, is equal and a being both numbers representing units of to the height through which this body must fall space; therefore, in the present case, we have in vacuo in order to acquire its terminal velocity For the whole resistance r, or MM,
in air. 2 a
The following description may render all these is supposed equal to the weight, or to g M; circumstances more easily conceived by some
readers. The terminal velocity is that where the and therefore r is equal to g, = 5
and 2 a= 2 a
resistance of the air balances and is equal to the ua
There is a consideration which ought to weight of the body. The resistance of the air to 8
any particular body is as the square of the velohave place here. A body descends in air, not city; therefore let' R be the whole resistance to by the whole of its weight, but by the excess of the body moving with the velocity 1, and r the its weight above that of the air which it displaces. resistance to its motion with the terminal velocity It descends by its specific gravity only as a stone
u; we must have r = R X u', and this must be does in water. Suppose a body thirty-two times W, the weight. Therefore, to obtain the terheavier than air, it will be buoyed up by a force minal velocity, divide the weight by the resist
1 equal to of its weight; and, instead of acquir
ance to the velocity 1, and the quotient is the 32
=u: and ing the velocity of thirty-two feet in a second, it square of the terminal velocity, or
R will only acquire a velocity of thirty-one, even this is a very expeditious method of determining though it sustained no resistance from the inertia it, if R be previously known. Then the comof the air. Let p be the weight of the body, and mon theorems give a, the fall necessary for pro
that of an equal bulk of air : the accelerative force of relative gravity on each particle will be ducing this velocity in vacuo = and the time
2 g' I.; and this relative accelerating force of the fall = = e, and eu = 2a, = the might be distinguished by another symbol y: space uniformly described with the velocity u But in all cases in which we have any interest, during the time of the fall, or its equal, the time and particularly in military projectilcs, of the extinction by the uniform action of the re
sistance r; and, since q extinguishes it in the small a quantity that it would be pedantic af- time e, R which is u2 times smaller will extinfectation to attend to it. is much more than guish it in the time u’e, and R will extinguish compensated when we make g = 32 feet, in- the velocity 1, which is u times less than u, in stead of 32, which it should be.
the time ue, that is, in the time 2 a; and the Let e be the time of this ascent in opposition body moving uniformly during the time 2 a = to gravity. The same theorems give us eu = 2 E, with the velocity 1, will describe the space a; and, since the resistance competent to this 2 a; and if the body begin to move with the veterminal velocity is equal to gravity, e will also locity 1, and be uniformly opposed by the rebe the time in which it would be extinguished sistance R, it will be brought to rest when it has by the uniform action of the resistance; for described the space a; and the space in which which reason we may call it the extinguishing the resistance to the velocity 1 will extinguish time for this velocity. Let R and E mark the that velocity by its uniform action is equal to resistance and extinguishing time for the same the height through which that body must fall in body moving with the velocity 1.
vacuo in order to acquire its terminal velocity in As the resistances are as the squares of the ve- air. And thus every thing is regulated by the